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Tokyo Probability Seminar
Seminar information archive ～10/20｜Next seminar｜Future seminars 10/21～
| Date, time & place | Monday 16:00 - 17:30 122Room #122 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | Makiko Sasada, Shuta Nakajima (Keio Univ.), Masato Hoshino (Science Tokyo), Masahisa Ebina (Science Tokyo) |
Future seminars
2026/11/02
16:00-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)
We are having teatime from 15:20 in the common room on the second floor. Please join us.
Koyo Oishi (Kyoto University)
Convergence rate estimates for semigroups and heat kernels associated with resistance forms
We are having teatime from 15:20 in the common room on the second floor. Please join us.
Koyo Oishi (Kyoto University)
Convergence rate estimates for semigroups and heat kernels associated with resistance forms
[ Abstract ]
In this talk, we present quantitative convergence rates for stochastic processes associated with resistance forms. While the qualitative convergence of heat kernels and semigroups under the Gromov-Hausdorff-vague convergence of underlying measured resistance metric spaces has been investigated previously, their quantitative convergence rates have remained unexplored. We establish explicit convergence rates for the associated semigroups and heat kernels under the assumptions of measure regularity and lower resistance estimates. Furthermore, we introduce a new metric that induces the Gromov-Hausdorff-vague topology and is convenient for obtaining quantitative estimates. As applications of our main results, we present several examples, including the Sierpinski gasket and the Bouchaud trap model. In some of these examples, we obtain the first quantitative estimates for previously known convergence results, while in others, we improve existing estimates.
In this talk, we present quantitative convergence rates for stochastic processes associated with resistance forms. While the qualitative convergence of heat kernels and semigroups under the Gromov-Hausdorff-vague convergence of underlying measured resistance metric spaces has been investigated previously, their quantitative convergence rates have remained unexplored. We establish explicit convergence rates for the associated semigroups and heat kernels under the assumptions of measure regularity and lower resistance estimates. Furthermore, we introduce a new metric that induces the Gromov-Hausdorff-vague topology and is convenient for obtaining quantitative estimates. As applications of our main results, we present several examples, including the Sierpinski gasket and the Bouchaud trap model. In some of these examples, we obtain the first quantitative estimates for previously known convergence results, while in others, we improve existing estimates.
2026/11/09
16:00-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)
We are having teatime from 15:20 in the common room on the second floor. Please join us.
Kenta Misumi (The University of Tokyo)
ディリクレ空間で、準至る所ベクトル束を作り、マルコフ過程に沿って準至る所ベクトルを平行移動する。
We are having teatime from 15:20 in the common room on the second floor. Please join us.
Kenta Misumi (The University of Tokyo)
ディリクレ空間で、準至る所ベクトル束を作り、マルコフ過程に沿って準至る所ベクトルを平行移動する。
[ Abstract ]
準正則 Dirichlet 空間上において準ベクトル束の概念を定式化することにより、その分類量として第一 Stiefel–Whitney 類が自然に現れる。さらに、この枠組みに適合する接続の理論を公理的に構築することで、対応する右過程の軌道に沿ったベクトルの準至る所での平行移動を構成することが可能となる。
この構成は、従来の Riemann 多様体あるいは sub-Riemann 多様体上の Brown 運動に沿う確率平行移動を包含するだけでなく、一般のジャンプ型過程に沿う確率平行移動も含む。さらには、RCD 空間など、特異空間上の Brown 運動に沿う確率平行移動さえも導く。
したがって本理論は、滑らかさに依存していた従来の平行移動の概念を、準正則 Dirichlet 形式と右過程を基礎とする非滑らかな対象へ拡張する一般的な枠組みを与えるものとなる。
準正則 Dirichlet 空間上において準ベクトル束の概念を定式化することにより、その分類量として第一 Stiefel–Whitney 類が自然に現れる。さらに、この枠組みに適合する接続の理論を公理的に構築することで、対応する右過程の軌道に沿ったベクトルの準至る所での平行移動を構成することが可能となる。
この構成は、従来の Riemann 多様体あるいは sub-Riemann 多様体上の Brown 運動に沿う確率平行移動を包含するだけでなく、一般のジャンプ型過程に沿う確率平行移動も含む。さらには、RCD 空間など、特異空間上の Brown 運動に沿う確率平行移動さえも導く。
したがって本理論は、滑らかさに依存していた従来の平行移動の概念を、準正則 Dirichlet 形式と右過程を基礎とする非滑らかな対象へ拡張する一般的な枠組みを与えるものとなる。
